Subgroups of the braid group \(B_ 4\) (Q1181612)

Let \(B_ n\) be the \((n-1)\)-generated braid group. It is well known that \(B_ n\) has unsolvable generalized word problem for \(n\geq 5\) and a solvable one for \(n\leq 3\), the answer for \(B_ 4\) is unknown. For \(n\geq 5\) the proof is based on the fact that in this case \(B_ n\) contains a subgroup isomorphic to the direct product \(F_ 2\times F_ 2\) of two free groups of rank two with unsolvable generalized word problem. The author shows that such a strategy fails for \(B_ 4\): it does not contain a subgroup isomorphic to \(F_ 2\times F_ 2\).